scholarly journals On the stability of the linear delay differential and difference equations

2001 ◽  
Vol 6 (5) ◽  
pp. 267-297 ◽  
Author(s):  
A. Ashyralyev ◽  
P. E. Sobolevskii
Keyword(s):  
Difference Equations ◽  
Initial Value Problem ◽  
Parabolic Type ◽  
Stability Estimates ◽  
Initial Value ◽  
Linear Delay ◽  
Accuracy Difference ◽  
The Stability ◽  
Delay Differential ◽  
Delay Partial Differential Equations

We consider the initial-value problem for linear delay partial differential equations of the parabolic type. We give a sufficient condition for the stability of the solution of this initial-value problem. We present the stability estimates for the solutions of the first and second order accuracy difference schemes for approximately solving this initial-value problem. We obtain the stability estimates in Hölder norms for the solutions of the initial-value problem of the delay differential and difference equations of the parabolic type.

scholarly journals A note on the difference schemes for hyperbolic equations

2001 ◽  
Vol 6 (2) ◽  
pp. 63-70 ◽  
Author(s):  
A. Ashyralyev ◽  
P. E. Sobolevskii
Keyword(s):  
Hilbert Space ◽  
Initial Value Problem ◽  
Hyperbolic Equations ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Initial Value ◽  
Order Accuracy ◽  
Accuracy Difference ◽  
The Difference ◽  
The Stability

The initial value problem for hyperbolic equationsd 2u(t)/dt 2+A u(t)=f(t)(0≤t≤1),u(0)=φ,u′(0)=ψ, in a Hilbert spaceHis considered. The first and second order accuracy difference schemes generated by the integer power ofAapproximately solving this initial value problem are presented. The stability estimates for the solution of these difference schemes are obtained.

scholarly journals On the stability of the Schrödinger equation with time delay

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 759-766 ◽  
Author(s):  
Deniz Agirseven
Keyword(s):  
Hilbert Space ◽  
Time Delay ◽  
Schrödinger Equation ◽  
Initial Value Problem ◽  
Schrodinger Equation ◽  
Stability Estimates ◽  
Initial Value ◽  
Dinger Equation ◽  
The Stability

In the present paper, the initial value problem for the Schr?dinger equation with time delay in a Hilbert space is investigated. Theorems on stability estimates for the solution of the problem are established. The applications of theorems for three types of Schr?dinger problems are provided.

scholarly journals On the Absolute Stable Difference Scheme for Third Order Delay Partial Differential Equations

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 1033
Author(s):  
Allaberen Ashyralyev ◽  
Evren Hınçal ◽  
Suleiman Ibrahim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Difference Scheme ◽  
Stability Estimates ◽  
Third Order ◽  
Partial Differential ◽  
The Absolute ◽  
The Stability ◽  
Delay Differential ◽  
Delay Partial Differential Equations

The initial value problem for the third order delay differential equation in a Hilbert space with an unbounded operator is investigated. The absolute stable three-step difference scheme of a first order of accuracy is constructed and analyzed. This difference scheme is built on the Taylor’s decomposition method on three and two points. The theorem on the stability of the presented difference scheme is proven. In practice, stability estimates for the solutions of three-step difference schemes for different types of delay partial differential equations are obtained. Finally, in order to ensure the coincidence between experimental and theoretical results and to clarify how efficient the proposed scheme is, some numerical experiments are tested.

scholarly journals Stability of a Second Order of Accuracy Difference Scheme for Hyperbolic Equation in a Hilbert Space

2007 ◽  
Vol 2007 ◽  
pp. 1-25 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Mehmet Emir Koksal
Keyword(s):  
Hilbert Space ◽  
Difference Scheme ◽  
Hyperbolic Equation ◽  
Initial Value Problem ◽  
Second Order ◽  
Stability Estimates ◽  
Initial Value ◽  
Order Of Accuracy ◽  
Positive Definite Operators ◽  
Accuracy Difference

The initial-value problem for hyperbolic equation d2u(t)/dt2+A(t)u(t)=f(t)(0≤t≤T), u(0)=ϕ,u′(0)=ψ in a Hilbert space H with the self-adjoint positive definite operators A(t) is considered. The second order of accuracy difference scheme for the approximately solving this initial-value problem is presented. The stability estimates for the solution of this difference scheme are established.

scholarly journals Quasilinearization of the Initial Value Problem for Difference Equations with “Maxima”

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
S. Hristova ◽  
A. Golev ◽  
K. Stefanova
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Initial Value Problem ◽  
Linear Difference Equation ◽  
Successive Approximations ◽  
Time Interval ◽  
Initial Value ◽  
Past Time ◽  
The Given ◽  
Unknown Solution

The object of investigation of the paper is a special type of difference equations containing the maximum value of the unknown function over a past time interval. These equations are adequate models of real processes which present state depends significantly on their maximal value over a past time interval. An algorithm based on the quasilinearization method is suggested to solve approximately the initial value problem for the given difference equation. Every successive approximation of the unknown solution is the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima,” and a formula for its explicit form is given. Also, each approximation is a lower/upper solution of the given mixed problem. It is proved the quadratic convergence of the successive approximations. The suggested algorithm is realized as a computer program, and it is applied to an example, illustrating the advantages of the suggested scheme.

scholarly journals The influence of circulation on the stability of vortices to mode-one disturbances

1995 ◽  
Vol 451 (1943) ◽  
pp. 747-755 ◽  
Keyword(s):  
Velocity Field ◽  
Angular Velocity ◽  
Initial Value Problem ◽  
Large Time ◽  
Asymptotic Methods ◽  
Two Dimensional ◽  
Initial Value ◽  
The Stability ◽  
Ultimate Fate ◽  
Solution Behaviour

The initial value problem for the two-dimensional inviscid vorticity equation, linearized about an azimuthal basic velocity field with monotonic angular velocity, is solved exactly for mode-one disturbances. The solution behaviour is investigated for large time using asymptotic methods. The circulation of the basic state is found to govern the ultimate fate of the disturbance: for basic state vorticity distributions with non-zero circulation, the perturbation tends to the steady solution first mentioned in Michalke & Timme (1967), while for zero circulation, the perturbation grows without bound. The latter case has potentially important implications for the stability of isolated eddies in geophysics.

scholarly journals Computer Simulation and Iterative Algorithm for Approximate Solving of Initial Value Problem for Riemann-Liouville Fractional Delay Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 477
Author(s):  
Snezhana Hristova ◽  
Kremena Stefanova ◽  
Angel Golev
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Finite Interval ◽  
Initial Value ◽  
Monotone Iterative ◽  
Practical Usefulness ◽  
Fractional Delay ◽  
Fractional Differential ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

The main aim of this paper is to suggest an algorithm for constructing two monotone sequences of mild lower and upper solutions which are convergent to the mild solution of the initial value problem for Riemann-Liouville fractional delay differential equation. The iterative scheme is based on a monotone iterative technique. The suggested scheme is computerized and applied to solve approximately the initial value problem for scalar nonlinear Riemann-Liouville fractional differential equations with a constant delay on a finite interval. The suggested and well-grounded algorithm is applied to a particular problem and the practical usefulness is illustrated.

Stability of time-dependent rotational Couette flow Part 2. Stability analysis

1971 ◽  
Vol 48 (2) ◽  
pp. 365-384 ◽  
Author(s):  
C. F. Chen ◽  
R. P. Kirchner
Keyword(s):  
Growth Rate ◽  
Stability Analysis ◽  
Kinetic Energy ◽  
Initial Value Problem ◽  
Basic Flow ◽  
The Other ◽  
Time Dependent ◽  
Stability Criteria ◽  
Initial Value ◽  
The Stability

The stability of the flow induced by an impulsively started inner cylinder in a Couette flow apparatus is investigated by using a linear stability analysis. Two approaches are taken; one is the treatment as an initial-value problem in which the time evolution of the initially distributed small random perturbations of given wavelength is monitored by numerically integrating the unsteady perturbation equations. The other is the quasi-steady approach, in which the stability of the instantaneous velocity profile of the basic flow is analyzed. With the quasi-steady approach, two stability criteria are investigated; one is the standard zero perturbation growth rate definition of stability, and the other is the momentary stability criterion in which the evolution of the basic flow velocity field is partially taken into account. In the initial-value problem approach, the predicted critical wavelengths agree remarkably well with those found experimentally. The kinetic energy of the perturbations decreases initially, reaches a minimum, then grows exponentially. By comparing with the experimental results, it may be concluded that when the perturbation kinetic energy has grown a thousand-fold, the secondary flow pattern is clearly visible. The time of intrinsic instability (the time at which perturbations first tend to grow) is about ¼ of the time required for a thousandfold increase, when the instability disks are clearly observable. With the quasi-steady approach, the critical times for marginal stability are comparable to those found using the initial-value problem approach. The predicted critical wavelengths, however, are about 1½ to 2 times larger than those observed. Both of these points are in agreement with the findings of Mahler, Schechter & Wissler (1968) treating the stability of a fluid layer with time-dependent density gradients. The zero growth rate and the momentary stability criteria give approximately the same results.

scholarly journals Existence, uniqueness and continuability of solutions of impulsive differential-difference equations

1999 ◽  
Vol 12 (3) ◽  
pp. 293-300 ◽  
Author(s):  
D. D. Bainov ◽  
I. M. Stamova
Keyword(s):  
Difference Equations ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Initial Value

We consider an initial value problem for impulsive differential-difference equations, and obtain sufficient conditions for the existence, uniqueness, and continuability of solutions of such problem.

Sign in / Sign up

Export Citation Format

Share Document